Category:Definitions/Contour Integrals

This category contains definitions related to Contour Integrals.
Related results can be found in Category:Contour Integrals.

Let $OA$ be a curve in a vector field $\mathbf F$.

Let $P$ be a point on $OA$.

Let $\d \mathbf l$ be a small element of length of $OA$ at $P$.

Let $\mathbf v$ be the vector induced by $\mathbf F$ on $P$.

Let $\mathbf v$ make an angle $\theta$ with the tangent to $OA$ at $P$.

Hence:

$\mathbf v \cdot \d \mathbf l = v \cos \theta \rd l$

where:

$\cdot$ denotes dot product

$v$ and $\d l$ denote the magnitude of $\mathbf v$ and $\d \mathbf l$ respectively.

The contour integral of $\mathbf v$ along $OA$ is therefore defined as:

$\ds \int_O^A \mathbf v \cdot \d \mathbf l = \int_O^A v \cos \theta \rd l$